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Full text of "Analytical Mechanics"

280                      ANALYTICAL  MKCHANICS
where m is the mass of the body and r the velocity of center of mass. But by the conservation of angular monu turn the angular momentum of the body about t ho axis af the blow must equal that of the blow itself. Therefore
where 7 is the moment of inertia of the body, a> its angu velocity and 6 the distance of the line of action of the bl from the axis.
Eliminating L between equations (U and (2) and solvi for Lf we obtain
. t                ho
// -,^ tnr>~    * b
//w   . 1 a>,                          (I
where a is the distance of the center of mass from the axis rotation. Equation (IX) gives the impulse* produced by t reaction of the axis.
If the blow is applied at a center of percussion // = Therefore
and                                6 =
PROBLIvMS,
1.   A square plate is moving on a .smooth horizontal plane with twc its sides parallel to the direction of motion.    Find the angular veloc with which it will rotate, also the impulsive reaction of the axis,
(a)  if one of the corners Is fixed;
(b)  if the middle point of one of the widen is fixed.
2.   An equilateral triangular plate in moving on a smooth homoi; plane in a direction perpendicular to ont* of its .sides.    Find the result angular velocity, ulao the impulse given by the axis,
(a)  if one of its cornera in fixed;
(b)  if the middle point of one of its sides is fixed.
3.   A hoop is moving on a smooth horizontal plane with its axis perp